The Noether-Lefschetz theorem and sums of 4 squares in the rational function field <span class="mathjax-formula">$R(x, y)$</span>

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C ^OMPOSITIO M ^ATHEMATICA

J.-L. C ^OLLIOT -T ^HÉLÈNE

The Noether-Lefschetz theorem and sums of 4 squares in the rational function field R(x, y)

Compositio Mathematica, tome 86, n^o2 (1993), p. 235-243

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The Noether-Lefschetz theorem and sums of 4 squares in the rational function field

R(x, y)

J.-L. COLLIOT-THÉLÈNE

C.N.R.S., ^URA^D.0752, Mathématiques, ^Bâtiment^425,Université de Paris-Sud, ^F-91405Orsay,

France

Received 2 March 1992; accepted ²⁸April ¹⁹⁹²

Résumé. Il _ya vingt ans, Cassels, ÊllisonêtPfister montrèrent _{que le}polynôme ^deMotzkin, ûn polynôme ^dedegré ^{6 de}Q[x, y], ^bienque positif ^sur^R2êt^donc^somme^de⁴carrés dans _{le corps}des fonctions R(x, y) (Hilbert), ^n’estpas ^sommede 3 carrés dans R(x, y). ^Dans^cettenote, je remarque

qu’une ^extension^d’un^théorèmeclassique ^de^MaxNoether, extension remontant à Lefschetz, permet de montrer l’existence en tout degré pair d ⁶^debeaucoup ^depolynômes ^deR[x, y] positifs ^sur^R2

mais non sommes de 3 carrés dans R(x, y).

Introduction

Let us first briefly recall the relevant background ^for^sumsof squares in the

polynomial ring R[x, y] ^{and in}the field of rational functions R(x, y) ^over^the^real

field R.

In 1888, ^Hilbert[12] proved that any nonnegative polynomial ^PⁱⁿR[x, y] ^of

total degree ât^most^{4 is}^{a sum}ôf3 squares of polynomials (a ^modernproof îs

not available; ⁱⁿ[5] ¹give ^ageometric proof for the weaker fact that P is a sum

of 3 squares in the function field R(x, y).) ^In^the^samearticle, ^Hilbert^also^showed

that a nonnegative polynomial ⁱⁿR[x, y] of (even) degree ⁶^need^not^be^{a sum}

of (any number) of squares of polynomials.

In 1893, ^Hilbert[13] showed that _anynonnegative polynomial ^PⁱⁿR[x, y]

nevertheless is a sum of squares in the function field R(x, y) ândâglance ât

Hilbert’s precise ^resulttogether ^with^anapplication of Euler’s multiplication

formula for sums of 4 _squaresreveals, ^as^Landau(1906) first noticed ([15], p.

282), ^that^Pactually ^is^{a sum}of 4 squares in R(x, y).

In 1967, ^Motzkinproduced ^anexplicit nonnegative polynomial ^P^EQ[x, y]

(of ^totaldegree 6) ^{which is}^not^{a sum}of any ^number^ofsquares in R[x, y]. ^More explicit examples ^were^latergiven by ^{R. M.}^Robinson.Choi, ^Lamand Reznick have since developed ^asystematic theory ^forcomparing semidefinite homog-

eneous forms with sums of squares of forms, ^andthey ^haveproduced examples

of higher degree positive polynomials ^which^are^not^sums^ofsquares of

polynomials.

In 1971, by techniques pertaining ^toelliptic curves, Cassels, ^Ellison^and

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236

Pfister [3] ^wereâble^to^{show that}the Motzkin polynomial îs^not^{a sum}ôf³

squares in the function field R(x, y). ^Thisproved ^that^the^so-calledPythagoras

number of R(x, y) (the minimum number of squares required ^toexpress ^an

arbitrary ^sum^ofsquares), ^which^was^known^to^be^at^most⁴by ^Landau’s^remark

to Hilbert’s 1893 paper, and to be at least equal ^to³(since ¹⁺^X2+ y2 ^is^not^a

sum of 2 squares in R(x, y), ^Cassels[2]), ^isactually equal ^to^{4. More}examples ^of

the same kind were later given by ^Christie[4].

Let us now turn to complex algebraic geometry. ^Aclassical theorem of Max Noether says that on a "general" surface of degree d ^{4 in}3-dimensional

projective space, all curves are cut out by ^another^surface(this ^{fails for}^d⁼2,3).

An equivalent ^statementis that the Picard group of X is spanned by the class of

a plane ^section.

Lefschetz, ^whogave ^newproofs of Noether’s theorem, also sketched a proof ([16], p. 359) ^{that for}the double cover X of projective plane p2, ^ramifiedalong â sufficiently "general" ^curveôfêvendegree d 6, the Picard _groupPic(X) ^{is free}

of rank _one,spanned by the inverse image ^of^a^lineⁱⁿ^theplane (this ^fails^for

d ⁼4). ^Variousproofs ^andextensions of these theorems have been obtained in

more recent years (Deligne [6], Griffiths-Harris [11] for Noether’s _case;

Steenbrink [18], ^Buium[1], ^Ein[9] ^for^moregeneral situations including

double covers). ^The^theorem^hasrecently ^been^usedby ^T.^Ford[10] ⁱⁿ^thestudy

of the Brauer group Br(X) ^of^such^a^double^cover^X.

In this note, ¹^remarkthat the precise ^versionof the Noether-Lefschetz theorem enables one to show:

THEOREM. For _anyeven degree d 6, ^there^exist(many) positive polynomials P(x, y) E R[x, y] of ^totaldegree d such that P is not a sum of 3 squares in R(x, y).

(The restriction to d 6, necessary in view of Hilbert’s result for d ⁼4,

mirrors the (necessary) geometric restriction in Lefschetz’s theorem.)

We thus get ^aradically ^newproof ^that^the^so-calledPythagoras ^{number of} R(x, y) ^{is 4.}^Theproof ^isconstructive, ^butproduces polynomials ^whose

coefficients are transcendental over Q, ⁱⁿ^contrastwith the Motzkin polynomial,

or the polynomials ⁱⁿ[4], whose coefficients lie in Q.

One may wonder whether ^asimilar approach might ^lead^to^someinformation

on the well-known _openproblem: ^{What is}^theêxactPythagoras ^number p(R(x, y, z)) ôfR(x, y, z)? Ît^{is known}^to^beât^least⁵(this ^follows^from^direct

combination of a result of Cassels ([2]; [14], p. 260) ^and^ofthe main result of

Cassels/Ellison/Pfister [3]). ^Thatp(R(x, y, z)) ^is^at^most⁸^was^shownby ^{Ax and}

Pfister. Is p(R(x, y, z)) equal ^to8? Lemma 1.2 below has an analogue ⁱⁿ^terms^of

the kernel of the _{map of}Galois cohomology H3(F, Z/2) -

H3(F(~-f),

^Z/2),

but there is no simple analogue of Lemma 1.1 at the H3 level (as M. Rost has

pointed out to me, ^onemay ^usealgebraic K-theory ^towrite down some

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analogous ^exactsequence, but the K-cohomology groups which this sequence involves look rather intractable.)

It was M. Ojanguren who insisted that one should find a new approach ^to^the Cassels/Ellison/Pfister result, via Lemma 1.2 below. The idea of using ^the

Noether-Lefschetz theorem came to me while 1 was reading ^apaper of T. Ford

[10] ^onthe Brauer group of double covers of the complex projective plane.

1 would like to express my hearty ^thanks^toProf. T.-Y. Lam for organizing ^the special year ônreal algebraic geometry ândquadratic ^formsâtBerkeley, ^1990-

1991, ^{and for}making ^our^visitpossible ^and^soenjoyable.

1. Some Galois cohomology.

LEMMA 1.1. Let k be a field, X/k ^be^a^smoothprojective geometrically integral variety. ^Letk(X) ^{be the}function field of ^{X. Let}^k^be^aseparable ^closureof k,

g ⁼

Gal(k/k)

^and^X⁼^XX k k. Then there is an exact sequence:

Proof. ^{This is}well-known, and easy ^todeduce from the exact sequence of Galois modules

(the map

k(X)* ~ Div(X)

^isthe divisor map from the multiplicative group of the function field

k(X)

^of^X^to^the^divisorgroup; that the kernel is k* is a

consequence of the hypothesis that X is proper and geometrically integral) together ^with^Hilbert’stheorem 90 and the following ^facts:

H1(g,

Div(X))

⁼⁰(since ^X^issmooth,

Div(X)

îs^{a sum}ôfpermutation modules, ândôneapplies Shapiro’s lemma);

Div(X)9

⁼Div(X);

H2(g,

k(X)*)

embeds into the Brauer group Br(k(X)) (restriction-inflation sequence).

(As â^matter^{of fact,}if smoothness is not assumed, ^theLeray spectral sequence for étale cohomology ^{and the}projection ^{X ~}Spec(k) yields ânêxactsequence

where Br(X) ⁼H2ét(X, Gm). Smoothness of X cornes in to guarantee

(Grothendieck) ^that^thenatural map Br(X) ~ Br(k(X)) ^is^aninjection.) ^D

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238

LEMMA 1.2. Let F be field, char(F) ~ ^{2. Let}f~F, f :0 ^0.^Thefollowing

conditions are equivalent:

(i) f is ^{a sum}of ^threesquares in F;

(ii) ( -1) ^is^{a sum}of ^twosquares ⁱⁿthe field

F(~-f);

(iii) ^thequaternion algebra ( -1, -1) ^is^trivial^over

F(~-f).

Proof. ^Theimplication (i) ^~(iii), ^whichîs^theonly ôneûsedbelow, îstrivial, (ii) ^~(iii) îswell-known, ând(ii) ^~(i) îsâneasy calculation. For a generalization,

see Lam [14], XI.2.6,2.7. 0

2. The Noether-Lefschetz theorem

1 shall use the following "generic" ^version^{of the}Noether-Lefschetz theorem,

which ^canbe read off from Lefschetz ([16] p. 359, ^areference 1 found in Buium’s

paper), Steenbrink [18], ^Buium[1], ^Ein[9]. (References ^forweighted projective

spaces ^are:Delorme [7], Dolgachev [8], ^Mori[17].)

THEOREM 2.1. Let m 3 be an integer, ^let^N⁼(m ⁺2)(m ⁺1)/2, ^let(ai)1iN

and (bij)1ijN ^beindependent variables. Let K ⁼Q(ai, bij) ^{be the}field spanned

over Q by these variables. Introduce the linear form

and the quadratic form

both defined ^over^{K. Let}^1t:^{X -}P2K ^{be the}^double^coverdefined ⁱⁿweighted projective space P(l, 1, 1, m)(x, y, ^tof weight ^{1 ;}^zof weight m) by ^theequation

where (a, b, c) ^runsthrough ^thetriples of nonnegative integers ^with^a⁺^b⁺^c⁼^m.

Then X is a smooth surface ôverK, ândfor ânyalgebraically ^closedfield ^L containing K, ^themap

is an isomorphism.

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3. Positive rational functions on the real affine plane ^which^are^not^sums^{of 3}

squares

Let m 3 be an integer ^and^N⁼(m ⁺2)(m ⁺1)/2. ^Let l(Tl, ... , TN) = 03A31iN aiTi ^be^alinear form and q(T1,..., TN) ⁼

03A31ijN bijTiTj

^a^quadratic

form, both with coefficients in C. Fixing ^anordering ^on^thepairs of nonnegative integers (a, b) ^{with a}⁺b m, ^wemay define a morphism ^{of affine}spaces:

Let P(x, Y)ER[x, y] ^{be the}polynomial

THEOREM 3.1. With notation as above, ^assume^{that the}coefficients (ai)1iN

and (bij)1ijN ^lieⁱⁿ^R.

(i) If ^thesecoefficients ^arealgebraically independent ^overthe rational field ^Q,

then the polynomial P(x, y) ^is^not^{a sum}of ³squares in the rational function field R(x, y).

(ii) ^There^exists^somereal number a > 0 such that if ^thecoefficients satisfy ^the inequalities

then the polynomial P(x, y) ^isstrictly positive ^on^R2.

Proof. ^Let^K⁼Q(ai, bij) ^{be the}^fieldspanned ^overQ by ^theai’s ^andbij’s. ^Let

x: X ~ p2 ^{be the}^double^coverdefined in weighted projective space P(1,1,1, m) (x, y, t ^ofweight 1; ^z^ofweight m) by ^theequation

where (a, b, c) ^runsthrough ^thetriples ^ofpositive integers ^{with a}^{+ b}⁺^c⁼^m.

The surface X and the morphism x ^are^defined^over^K.

The surface X appears ^morenaturally ^{as an}intersection of two varieties in

projective ^spacePNK. ^Let^indeed^Y^be^theimage ^{of the}weighted projective space

P(1, 1, 1, m) ⁱⁿprojective space

PNQ

^{under the}embedding given by ^{forms of} degree m, ^i.e.

for some fixed ordering of all natural integers ^a,b, ^csatisfying a ⁺^b⁺^c⁼^m.

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240

The variety ^Y^is^a^smoothprojective ^threefoldⁱⁿ

PNQ.

Our surface X ⁼X"q ^is^noneother than the intersection YK ⁿQ c P) ^of^the quadric Q c P) given by ^theequation

in P’ with the threefold YK ⁼Y x QK. Projection ^1t^{from the}point (o, o, o,1)

makes X into a double cover of the usual projective ^spacepi, ^ramifiedalong ^the

curve given by ^theequation

Since the quadric Q ^isgeneric, ^astraightforward application of Bertini’s theorem implies ^{that X is}geometrically connected and smooth over K.

Since the coefficients (ai)1iN ^and(bij)1ijN ^arealgebraically independent

over Q, ^{and m}3, ^theNoether-Lefschetz theorem 2.1 implies that the map

is an isomorphism. ^Since^thecoefficients (ai)1iN ^and(bij)1ijN ^{lie in}^R,^we

may consider the surface XR. ^Fromthe obvious commutative diagram

where the left vertical _{map is}identity ^onZ, ^we^conclude^that^theinjective map

is an isomorphism.

Let R(X) be the function field of XR. According ^to^Lemma1.1, ^we^{have the}

exact sequence

and we conclude that the map Br(R) ~ Br(R(X)) ^is^aninjection: the Hamilton

quaternion algebra ^does^notsplit ^over^the^function^fieldR(X).

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Now the field R(X) ^isR-isomorphic ^to^{the field}R(x,

y)(~-

^P).^Indeed,ⁱⁿ^affine

coordinates (we ^{let t}⁼1), ^theequation ^of^X^reads

i.e.

and an obvious change of variables realizes an isomorphism with the affine surface with equation

From Lemma 1.2 we conclude that the polynomial P(x, y) ^is^not^{a sum}^{of three}

squares in the field R(x, y), ^{which is}^claim(i).

Let us now prove (ii). ^Thepolynomial P(x, y) ^iscertainly positive ^on^R2^{if the} quadratic ^form

is positive ^definite.^Letqo(Tl, ... , TN) ⁼03A31iN T2i and 10(Tl’...’ TN) ⁼^{0. If}^Bⁱⁿ (ii) ^is^chosen^smallenough, the coefficients of the quadratic ^form4q - l2 ^{will be}

very close to those of 4qo - 12. ^Since^this^last^formclearly ^ispositive definite, ^so

will be 4q - ^12.

The proof ôfôurtheorem is now complete. However, it remains to observe that the theorem is not empty. ^Forthis, ^wesimply ^note^{that for}ânynon-empty open interval 7 of the real line R, ^thefield extension of Q generated by ^the

elements of I is R, ^hencecertainly of infinite transcendence degree ^overQ. ^The

existence of suitable (ai) ^and(bij) is thus clear. D

4. Remarks

4.1. The above proof gives ^amethod for producing explicit polynomials P(x, y)

which are positive ^but^not^sumsôf^threesquares in R(x, y) : ^{it is}enough ^to produce sufficiently many algebraically independent elements in R. Whether such elements ^canbe "explicitly" produced îsâ^matterôf^taste.

4.2. The same proof would work if R was replaced by ânarchimedean real closed field R of infinite transcendence degree ôverQ (the restrictions on R being imposed ⁱⁿôrder^toênsure^thatthe theorem is not empty.)

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242

4.3. In some of the published variants of the Noether-Lefschetz theorem, ^rather

than using generic coefficients for the equations ^athand, ^one^deals^with

"general" coefficients.

The published proofs ^show:În^thespace Z of coefficients, ^{which is}^some Zariski _openset in projective space ôverthe complex field, ^thereêxistsâ countable union of closed analytic subvarieties Zi(i~N), Zi ~ Z, such that for

z e Z(C) away from these varieties, the Picard _groupof the surface Xz ^is^reduced

to Z.

(I ^amtold that one may actually ^take^theZi ^to^be^closedalgebraic subvarieties.)

Granting this result in the context of double _covers,it is possible ^togive ^a

variant of the proof of Theorem 3.1. To prove that the theorem thus obtained is not empty, ^rather^thanusing âtranscendence degree argument, ône^hereûses Baire’s category ^theorem.

4.4. The polynomials produced by Cassels/Ellison/Pfister [3] ^and^Christie[4]

lie in Q[x, y]. ^Thepolynomials P(x, y) ^we^exhibitcertainly ^do^not^lieⁱⁿQ[x, y],

since their coefficients are transcendental. 1 wonder whether a combination of my approach ^{with the}technique ûsedby ^Terasoma[19] (itself âcombination of the Noether method as expounded by Deligne ândôf^Hilbert’sirreducibility theorem) ^would^lead^topolynomials ⁱⁿQ[x, y].

References

1. A. Buium: Sur le nombre de Picard des revêtements doubles des surfaces algébriques, ^C.R.^Acad.

Sc. Paris 296 (1983) ^SérieI, ^361-364.

2. J. W. S. Cassels: On the representation of rational functions as sums of squares, Acta Arithmetica 9 (1964), ^79-82.

3. J. W. S. Cassels, W. Ellison and A. Pfister: On sums of squares and ^onelliptic curves over

function fields, ^J.^NumberTheory ³(1971), ^125-149.

4. M. R. Christie: Positive definite rational functions in two variables which are not the sum of three squares, ^J.Number Theory ⁸(1976), ^224-232.

5. J.-L. Colliot-Thélène: Real rational surfaces without a real point, Archiv der Mathematik. 58 (1992) ^392-396.

6. P. Deligne: ^Lethéorème de Noether, ^{in SGA}7 II, exp. XIX, Springer L.N.M. 340 (1973), ^328-

340.

7. C. Delorme: Espaces projectifs anisotropes, ^Bull.^Soc.^Math.^France¹⁰³(1975), ^203-223.

8. I. Dolgachev: Weighted projective varieties, ⁱⁿSpringer ^L.N.M.⁹⁵⁶(1982), ^34-71.

9. L. Ein: An analogue of Max Noether’s theorem, Duke Mathematical Journal 52 (1985),

689-706.

10. T. Ford: The Brauer group and ramified double covers of surfaces, preprint ^1991.

11. P. Griffiths and J. Harris: On the Noether-Lefschetz theorem and some remarks on

codimension-two cycles, ^Math.^Ann.²⁷¹(1985), ^31-51.

12. D. Hilbert: Ueber die Darstellung definiter Formen als Summe von Formenquadraten, ^Math.

Ann. 42 (1888), ^342-350.

13. D. Hilbert: Ueber ternäre definite Formen, Acta Math. 17 (1893), ^169-197.

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14. T.-Y. Lam: The algebraic theory of quadratic forms, Benjamin/Cummings ^1973.

15. E. Landau: Ueber die Darstellung definiter Funktionen durch Quadrate, Math. Ann. 62 (1906),

272-285.

16. S. Lefschetz: On certain numerical invariants of algebraic varieties with application ^to^Abelian varieties, ^Trans.Amer. Math. Soc. 22 (1921), ^327-482.

17. S. Mori: On a generalisation ^ofcomplete intersections, ^J.of Math. of Kyoto University ¹⁵(1975), 619-646.

18. J. Steenbrink: On the Picard group of certain smooth surfaces in weighted projective space, in Algebraic geometry, Proceedings, ^LaRabida, 1981, Springer ^L.N.M.⁹⁶¹(1982), ^302-313.

19. T. Terasoma: Complete intersections with middle Picard number 1 defined over Q, ^Math.^{Z. 189} (1985), ^289-296.